Properties

Label 570.8.a.b.1.3
Level $570$
Weight $8$
Character 570.1
Self dual yes
Analytic conductor $178.059$
Analytic rank $1$
Dimension $4$
CM no
Inner twists $1$

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Newspace parameters

Level: \( N \) \(=\) \( 570 = 2 \cdot 3 \cdot 5 \cdot 19 \)
Weight: \( k \) \(=\) \( 8 \)
Character orbit: \([\chi]\) \(=\) 570.a (trivial)

Newform invariants

Self dual: yes
Analytic conductor: \(178.059464526\)
Analytic rank: \(1\)
Dimension: \(4\)
Coefficient field: \(\mathbb{Q}[x]/(x^{4} - \cdots)\)
Defining polynomial: \(x^{4} - x^{3} - 3046 x^{2} + 50476 x + 497070\)
Coefficient ring: \(\Z[a_1, \ldots, a_{13}]\)
Coefficient ring index: \( 2^{5}\cdot 3 \)
Twist minimal: yes
Fricke sign: \(-1\)
Sato-Tate group: $\mathrm{SU}(2)$

Embedding invariants

Embedding label 1.3
Root \(36.0320\) of defining polynomial
Character \(\chi\) \(=\) 570.1

$q$-expansion

\(f(q)\) \(=\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +216.000 q^{6} +218.883 q^{7} +512.000 q^{8} +729.000 q^{9} +O(q^{10})\) \(q+8.00000 q^{2} +27.0000 q^{3} +64.0000 q^{4} -125.000 q^{5} +216.000 q^{6} +218.883 q^{7} +512.000 q^{8} +729.000 q^{9} -1000.00 q^{10} +2680.26 q^{11} +1728.00 q^{12} -12154.8 q^{13} +1751.06 q^{14} -3375.00 q^{15} +4096.00 q^{16} -17716.1 q^{17} +5832.00 q^{18} +6859.00 q^{19} -8000.00 q^{20} +5909.83 q^{21} +21442.1 q^{22} +61090.5 q^{23} +13824.0 q^{24} +15625.0 q^{25} -97238.2 q^{26} +19683.0 q^{27} +14008.5 q^{28} -50427.7 q^{29} -27000.0 q^{30} +18940.3 q^{31} +32768.0 q^{32} +72367.1 q^{33} -141729. q^{34} -27360.3 q^{35} +46656.0 q^{36} -375078. q^{37} +54872.0 q^{38} -328179. q^{39} -64000.0 q^{40} +16560.1 q^{41} +47278.6 q^{42} -9999.12 q^{43} +171537. q^{44} -91125.0 q^{45} +488724. q^{46} -931134. q^{47} +110592. q^{48} -775633. q^{49} +125000. q^{50} -478334. q^{51} -777905. q^{52} -1.38828e6 q^{53} +157464. q^{54} -335033. q^{55} +112068. q^{56} +185193. q^{57} -403421. q^{58} +1.24289e6 q^{59} -216000. q^{60} +1.98086e6 q^{61} +151522. q^{62} +159565. q^{63} +262144. q^{64} +1.51935e6 q^{65} +578937. q^{66} +1.61129e6 q^{67} -1.13383e6 q^{68} +1.64944e6 q^{69} -218883. q^{70} -5.75895e6 q^{71} +373248. q^{72} +1.60335e6 q^{73} -3.00063e6 q^{74} +421875. q^{75} +438976. q^{76} +586663. q^{77} -2.62543e6 q^{78} -2.22482e6 q^{79} -512000. q^{80} +531441. q^{81} +132481. q^{82} -1.73453e6 q^{83} +378229. q^{84} +2.21451e6 q^{85} -79993.0 q^{86} -1.36155e6 q^{87} +1.37229e6 q^{88} -1.31072e6 q^{89} -729000. q^{90} -2.66047e6 q^{91} +3.90979e6 q^{92} +511388. q^{93} -7.44907e6 q^{94} -857375. q^{95} +884736. q^{96} +8.90120e6 q^{97} -6.20507e6 q^{98} +1.95391e6 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} - 500q^{5} + 864q^{6} - 742q^{7} + 2048q^{8} + 2916q^{9} + O(q^{10}) \) \( 4q + 32q^{2} + 108q^{3} + 256q^{4} - 500q^{5} + 864q^{6} - 742q^{7} + 2048q^{8} + 2916q^{9} - 4000q^{10} - 354q^{11} + 6912q^{12} - 6366q^{13} - 5936q^{14} - 13500q^{15} + 16384q^{16} - 16412q^{17} + 23328q^{18} + 27436q^{19} - 32000q^{20} - 20034q^{21} - 2832q^{22} - 68140q^{23} + 55296q^{24} + 62500q^{25} - 50928q^{26} + 78732q^{27} - 47488q^{28} - 120486q^{29} - 108000q^{30} - 223328q^{31} + 131072q^{32} - 9558q^{33} - 131296q^{34} + 92750q^{35} + 186624q^{36} - 409930q^{37} + 219488q^{38} - 171882q^{39} - 256000q^{40} + 209182q^{41} - 160272q^{42} - 983566q^{43} - 22656q^{44} - 364500q^{45} - 545120q^{46} - 371420q^{47} + 442368q^{48} - 832632q^{49} + 500000q^{50} - 443124q^{51} - 407424q^{52} - 1254692q^{53} + 629856q^{54} + 44250q^{55} - 379904q^{56} + 740772q^{57} - 963888q^{58} - 797084q^{59} - 864000q^{60} - 3424652q^{61} - 1786624q^{62} - 540918q^{63} + 1048576q^{64} + 795750q^{65} - 76464q^{66} - 1072972q^{67} - 1050368q^{68} - 1839780q^{69} + 742000q^{70} - 2077240q^{71} + 1492992q^{72} - 257780q^{73} - 3279440q^{74} + 1687500q^{75} + 1755904q^{76} - 2436036q^{77} - 1375056q^{78} - 2112232q^{79} - 2048000q^{80} + 2125764q^{81} + 1673456q^{82} - 8743304q^{83} - 1282176q^{84} + 2051500q^{85} - 7868528q^{86} - 3253122q^{87} - 181248q^{88} - 18352170q^{89} - 2916000q^{90} - 7018432q^{91} - 4360960q^{92} - 6029856q^{93} - 2971360q^{94} - 3429500q^{95} + 3538944q^{96} + 18150q^{97} - 6661056q^{98} - 258066q^{99} + O(q^{100}) \)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))
Significant digits:
\(n\) \(a_n\) \(a_n / n^{(k-1)/2}\) \( \alpha_n \) \( \theta_n \)
\(p\) \(a_p\) \(a_p / p^{(k-1)/2}\) \( \alpha_p\) \( \theta_p \)
\(2\) 8.00000 0.707107
\(3\) 27.0000 0.577350
\(4\) 64.0000 0.500000
\(5\) −125.000 −0.447214
\(6\) 216.000 0.408248
\(7\) 218.883 0.241195 0.120597 0.992701i \(-0.461519\pi\)
0.120597 + 0.992701i \(0.461519\pi\)
\(8\) 512.000 0.353553
\(9\) 729.000 0.333333
\(10\) −1000.00 −0.316228
\(11\) 2680.26 0.607160 0.303580 0.952806i \(-0.401818\pi\)
0.303580 + 0.952806i \(0.401818\pi\)
\(12\) 1728.00 0.288675
\(13\) −12154.8 −1.53442 −0.767211 0.641394i \(-0.778356\pi\)
−0.767211 + 0.641394i \(0.778356\pi\)
\(14\) 1751.06 0.170550
\(15\) −3375.00 −0.258199
\(16\) 4096.00 0.250000
\(17\) −17716.1 −0.874573 −0.437286 0.899322i \(-0.644060\pi\)
−0.437286 + 0.899322i \(0.644060\pi\)
\(18\) 5832.00 0.235702
\(19\) 6859.00 0.229416
\(20\) −8000.00 −0.223607
\(21\) 5909.83 0.139254
\(22\) 21442.1 0.429327
\(23\) 61090.5 1.04695 0.523475 0.852041i \(-0.324635\pi\)
0.523475 + 0.852041i \(0.324635\pi\)
\(24\) 13824.0 0.204124
\(25\) 15625.0 0.200000
\(26\) −97238.2 −1.08500
\(27\) 19683.0 0.192450
\(28\) 14008.5 0.120597
\(29\) −50427.7 −0.383951 −0.191976 0.981400i \(-0.561489\pi\)
−0.191976 + 0.981400i \(0.561489\pi\)
\(30\) −27000.0 −0.182574
\(31\) 18940.3 0.114188 0.0570940 0.998369i \(-0.481817\pi\)
0.0570940 + 0.998369i \(0.481817\pi\)
\(32\) 32768.0 0.176777
\(33\) 72367.1 0.350544
\(34\) −141729. −0.618416
\(35\) −27360.3 −0.107866
\(36\) 46656.0 0.166667
\(37\) −375078. −1.21735 −0.608675 0.793419i \(-0.708299\pi\)
−0.608675 + 0.793419i \(0.708299\pi\)
\(38\) 54872.0 0.162221
\(39\) −328179. −0.885899
\(40\) −64000.0 −0.158114
\(41\) 16560.1 0.0375249 0.0187624 0.999824i \(-0.494027\pi\)
0.0187624 + 0.999824i \(0.494027\pi\)
\(42\) 47278.6 0.0984674
\(43\) −9999.12 −0.0191788 −0.00958942 0.999954i \(-0.503052\pi\)
−0.00958942 + 0.999954i \(0.503052\pi\)
\(44\) 171537. 0.303580
\(45\) −91125.0 −0.149071
\(46\) 488724. 0.740306
\(47\) −931134. −1.30819 −0.654093 0.756414i \(-0.726950\pi\)
−0.654093 + 0.756414i \(0.726950\pi\)
\(48\) 110592. 0.144338
\(49\) −775633. −0.941825
\(50\) 125000. 0.141421
\(51\) −478334. −0.504935
\(52\) −777905. −0.767211
\(53\) −1.38828e6 −1.28088 −0.640442 0.768006i \(-0.721249\pi\)
−0.640442 + 0.768006i \(0.721249\pi\)
\(54\) 157464. 0.136083
\(55\) −335033. −0.271530
\(56\) 112068. 0.0852752
\(57\) 185193. 0.132453
\(58\) −403421. −0.271494
\(59\) 1.24289e6 0.787865 0.393932 0.919139i \(-0.371114\pi\)
0.393932 + 0.919139i \(0.371114\pi\)
\(60\) −216000. −0.129099
\(61\) 1.98086e6 1.11738 0.558689 0.829377i \(-0.311305\pi\)
0.558689 + 0.829377i \(0.311305\pi\)
\(62\) 151522. 0.0807432
\(63\) 159565. 0.0803983
\(64\) 262144. 0.125000
\(65\) 1.51935e6 0.686215
\(66\) 578937. 0.247872
\(67\) 1.61129e6 0.654502 0.327251 0.944938i \(-0.393878\pi\)
0.327251 + 0.944938i \(0.393878\pi\)
\(68\) −1.13383e6 −0.437286
\(69\) 1.64944e6 0.604457
\(70\) −218883. −0.0762725
\(71\) −5.75895e6 −1.90959 −0.954793 0.297272i \(-0.903923\pi\)
−0.954793 + 0.297272i \(0.903923\pi\)
\(72\) 373248. 0.117851
\(73\) 1.60335e6 0.482391 0.241195 0.970477i \(-0.422461\pi\)
0.241195 + 0.970477i \(0.422461\pi\)
\(74\) −3.00063e6 −0.860797
\(75\) 421875. 0.115470
\(76\) 438976. 0.114708
\(77\) 586663. 0.146444
\(78\) −2.62543e6 −0.626425
\(79\) −2.22482e6 −0.507692 −0.253846 0.967245i \(-0.581696\pi\)
−0.253846 + 0.967245i \(0.581696\pi\)
\(80\) −512000. −0.111803
\(81\) 531441. 0.111111
\(82\) 132481. 0.0265341
\(83\) −1.73453e6 −0.332973 −0.166486 0.986044i \(-0.553242\pi\)
−0.166486 + 0.986044i \(0.553242\pi\)
\(84\) 378229. 0.0696269
\(85\) 2.21451e6 0.391121
\(86\) −79993.0 −0.0135615
\(87\) −1.36155e6 −0.221674
\(88\) 1.37229e6 0.214663
\(89\) −1.31072e6 −0.197081 −0.0985407 0.995133i \(-0.531417\pi\)
−0.0985407 + 0.995133i \(0.531417\pi\)
\(90\) −729000. −0.105409
\(91\) −2.66047e6 −0.370095
\(92\) 3.90979e6 0.523475
\(93\) 511388. 0.0659265
\(94\) −7.44907e6 −0.925028
\(95\) −857375. −0.102598
\(96\) 884736. 0.102062
\(97\) 8.90120e6 0.990256 0.495128 0.868820i \(-0.335121\pi\)
0.495128 + 0.868820i \(0.335121\pi\)
\(98\) −6.20507e6 −0.665971
\(99\) 1.95391e6 0.202387
\(100\) 1.00000e6 0.100000
\(101\) −1.89951e7 −1.83450 −0.917249 0.398315i \(-0.869595\pi\)
−0.917249 + 0.398315i \(0.869595\pi\)
\(102\) −3.82667e6 −0.357043
\(103\) 1.05914e6 0.0955042 0.0477521 0.998859i \(-0.484794\pi\)
0.0477521 + 0.998859i \(0.484794\pi\)
\(104\) −6.22324e6 −0.542500
\(105\) −738729. −0.0622762
\(106\) −1.11062e7 −0.905722
\(107\) −7.98725e6 −0.630310 −0.315155 0.949040i \(-0.602057\pi\)
−0.315155 + 0.949040i \(0.602057\pi\)
\(108\) 1.25971e6 0.0962250
\(109\) −9.55047e6 −0.706369 −0.353185 0.935554i \(-0.614901\pi\)
−0.353185 + 0.935554i \(0.614901\pi\)
\(110\) −2.68026e6 −0.192001
\(111\) −1.01271e7 −0.702838
\(112\) 896543. 0.0602987
\(113\) 1.67736e7 1.09358 0.546790 0.837270i \(-0.315850\pi\)
0.546790 + 0.837270i \(0.315850\pi\)
\(114\) 1.48154e6 0.0936586
\(115\) −7.63631e6 −0.468211
\(116\) −3.22737e6 −0.191976
\(117\) −8.86083e6 −0.511474
\(118\) 9.94315e6 0.557105
\(119\) −3.87774e6 −0.210942
\(120\) −1.72800e6 −0.0912871
\(121\) −1.23034e7 −0.631357
\(122\) 1.58469e7 0.790106
\(123\) 447123. 0.0216650
\(124\) 1.21218e6 0.0570940
\(125\) −1.95312e6 −0.0894427
\(126\) 1.27652e6 0.0568502
\(127\) −1.47405e7 −0.638558 −0.319279 0.947661i \(-0.603441\pi\)
−0.319279 + 0.947661i \(0.603441\pi\)
\(128\) 2.09715e6 0.0883883
\(129\) −269976. −0.0110729
\(130\) 1.21548e7 0.485227
\(131\) −1.05608e6 −0.0410436 −0.0205218 0.999789i \(-0.506533\pi\)
−0.0205218 + 0.999789i \(0.506533\pi\)
\(132\) 4.63150e6 0.175272
\(133\) 1.50132e6 0.0553339
\(134\) 1.28903e7 0.462802
\(135\) −2.46038e6 −0.0860663
\(136\) −9.07063e6 −0.309208
\(137\) 9.05173e6 0.300753 0.150376 0.988629i \(-0.451951\pi\)
0.150376 + 0.988629i \(0.451951\pi\)
\(138\) 1.31955e7 0.427416
\(139\) −4.40231e7 −1.39037 −0.695183 0.718833i \(-0.744676\pi\)
−0.695183 + 0.718833i \(0.744676\pi\)
\(140\) −1.75106e6 −0.0539328
\(141\) −2.51406e7 −0.755282
\(142\) −4.60716e7 −1.35028
\(143\) −3.25780e7 −0.931640
\(144\) 2.98598e6 0.0833333
\(145\) 6.30346e6 0.171708
\(146\) 1.28268e7 0.341102
\(147\) −2.09421e7 −0.543763
\(148\) −2.40050e7 −0.608675
\(149\) 2.40592e7 0.595839 0.297920 0.954591i \(-0.403707\pi\)
0.297920 + 0.954591i \(0.403707\pi\)
\(150\) 3.37500e6 0.0816497
\(151\) −4.28905e7 −1.01378 −0.506888 0.862012i \(-0.669204\pi\)
−0.506888 + 0.862012i \(0.669204\pi\)
\(152\) 3.51181e6 0.0811107
\(153\) −1.29150e7 −0.291524
\(154\) 4.69330e6 0.103551
\(155\) −2.36754e6 −0.0510665
\(156\) −2.10034e7 −0.442950
\(157\) 2.54178e7 0.524191 0.262096 0.965042i \(-0.415586\pi\)
0.262096 + 0.965042i \(0.415586\pi\)
\(158\) −1.77986e7 −0.358993
\(159\) −3.74834e7 −0.739519
\(160\) −4.09600e6 −0.0790569
\(161\) 1.33716e7 0.252519
\(162\) 4.25153e6 0.0785674
\(163\) 1.31342e7 0.237546 0.118773 0.992921i \(-0.462104\pi\)
0.118773 + 0.992921i \(0.462104\pi\)
\(164\) 1.05985e6 0.0187624
\(165\) −9.04589e6 −0.156768
\(166\) −1.38762e7 −0.235447
\(167\) 1.66839e7 0.277198 0.138599 0.990349i \(-0.455740\pi\)
0.138599 + 0.990349i \(0.455740\pi\)
\(168\) 3.02583e6 0.0492337
\(169\) 8.49899e7 1.35445
\(170\) 1.77161e7 0.276564
\(171\) 5.00021e6 0.0764719
\(172\) −639944. −0.00958942
\(173\) −3.14081e7 −0.461191 −0.230595 0.973050i \(-0.574067\pi\)
−0.230595 + 0.973050i \(0.574067\pi\)
\(174\) −1.08924e7 −0.156747
\(175\) 3.42004e6 0.0482390
\(176\) 1.09784e7 0.151790
\(177\) 3.35581e7 0.454874
\(178\) −1.04858e7 −0.139358
\(179\) −3.56579e7 −0.464698 −0.232349 0.972633i \(-0.574641\pi\)
−0.232349 + 0.972633i \(0.574641\pi\)
\(180\) −5.83200e6 −0.0745356
\(181\) −8.86947e7 −1.11179 −0.555895 0.831253i \(-0.687624\pi\)
−0.555895 + 0.831253i \(0.687624\pi\)
\(182\) −2.12837e7 −0.261697
\(183\) 5.34833e7 0.645119
\(184\) 3.12783e7 0.370153
\(185\) 4.68848e7 0.544416
\(186\) 4.09110e6 0.0466171
\(187\) −4.74837e7 −0.531006
\(188\) −5.95926e7 −0.654093
\(189\) 4.30827e6 0.0464180
\(190\) −6.85900e6 −0.0725476
\(191\) 4.33253e7 0.449909 0.224954 0.974369i \(-0.427777\pi\)
0.224954 + 0.974369i \(0.427777\pi\)
\(192\) 7.07789e6 0.0721688
\(193\) −3.27194e7 −0.327608 −0.163804 0.986493i \(-0.552377\pi\)
−0.163804 + 0.986493i \(0.552377\pi\)
\(194\) 7.12096e7 0.700216
\(195\) 4.10223e7 0.396186
\(196\) −4.96405e7 −0.470913
\(197\) 2.52473e7 0.235279 0.117639 0.993056i \(-0.462467\pi\)
0.117639 + 0.993056i \(0.462467\pi\)
\(198\) 1.56313e7 0.143109
\(199\) −1.77828e8 −1.59961 −0.799806 0.600259i \(-0.795064\pi\)
−0.799806 + 0.600259i \(0.795064\pi\)
\(200\) 8.00000e6 0.0707107
\(201\) 4.35047e7 0.377877
\(202\) −1.51961e8 −1.29719
\(203\) −1.10377e7 −0.0926070
\(204\) −3.06134e7 −0.252467
\(205\) −2.07001e6 −0.0167816
\(206\) 8.47311e6 0.0675317
\(207\) 4.45350e7 0.348984
\(208\) −4.97859e7 −0.383606
\(209\) 1.83839e7 0.139292
\(210\) −5.90983e6 −0.0440359
\(211\) −4.30155e7 −0.315236 −0.157618 0.987500i \(-0.550382\pi\)
−0.157618 + 0.987500i \(0.550382\pi\)
\(212\) −8.88496e7 −0.640442
\(213\) −1.55492e8 −1.10250
\(214\) −6.38980e7 −0.445696
\(215\) 1.24989e6 0.00857704
\(216\) 1.00777e7 0.0680414
\(217\) 4.14570e6 0.0275416
\(218\) −7.64037e7 −0.499479
\(219\) 4.32905e7 0.278508
\(220\) −2.14421e7 −0.135765
\(221\) 2.15335e8 1.34196
\(222\) −8.10169e7 −0.496981
\(223\) 1.28014e8 0.773019 0.386510 0.922285i \(-0.373681\pi\)
0.386510 + 0.922285i \(0.373681\pi\)
\(224\) 7.17234e6 0.0426376
\(225\) 1.13906e7 0.0666667
\(226\) 1.34189e8 0.773278
\(227\) 8.82730e7 0.500884 0.250442 0.968132i \(-0.419424\pi\)
0.250442 + 0.968132i \(0.419424\pi\)
\(228\) 1.18524e7 0.0662266
\(229\) 2.34752e6 0.0129177 0.00645885 0.999979i \(-0.497944\pi\)
0.00645885 + 0.999979i \(0.497944\pi\)
\(230\) −6.10905e7 −0.331075
\(231\) 1.58399e7 0.0845494
\(232\) −2.58190e7 −0.135747
\(233\) −2.06062e8 −1.06722 −0.533609 0.845731i \(-0.679165\pi\)
−0.533609 + 0.845731i \(0.679165\pi\)
\(234\) −7.08866e7 −0.361667
\(235\) 1.16392e8 0.585039
\(236\) 7.95452e7 0.393932
\(237\) −6.00702e7 −0.293116
\(238\) −3.10219e7 −0.149159
\(239\) −1.44459e8 −0.684466 −0.342233 0.939615i \(-0.611183\pi\)
−0.342233 + 0.939615i \(0.611183\pi\)
\(240\) −1.38240e7 −0.0645497
\(241\) 1.38118e8 0.635608 0.317804 0.948156i \(-0.397055\pi\)
0.317804 + 0.948156i \(0.397055\pi\)
\(242\) −9.84269e7 −0.446437
\(243\) 1.43489e7 0.0641500
\(244\) 1.26775e8 0.558689
\(245\) 9.69542e7 0.421197
\(246\) 3.57698e6 0.0153195
\(247\) −8.33696e7 −0.352021
\(248\) 9.69743e6 0.0403716
\(249\) −4.68323e7 −0.192242
\(250\) −1.56250e7 −0.0632456
\(251\) −1.44952e8 −0.578584 −0.289292 0.957241i \(-0.593420\pi\)
−0.289292 + 0.957241i \(0.593420\pi\)
\(252\) 1.02122e7 0.0401991
\(253\) 1.63739e8 0.635667
\(254\) −1.17924e8 −0.451529
\(255\) 5.97917e7 0.225814
\(256\) 1.67772e7 0.0625000
\(257\) 5.91829e7 0.217486 0.108743 0.994070i \(-0.465318\pi\)
0.108743 + 0.994070i \(0.465318\pi\)
\(258\) −2.15981e6 −0.00782973
\(259\) −8.20981e7 −0.293619
\(260\) 9.72382e7 0.343107
\(261\) −3.67618e7 −0.127984
\(262\) −8.44861e6 −0.0290222
\(263\) −2.51555e8 −0.852685 −0.426343 0.904562i \(-0.640198\pi\)
−0.426343 + 0.904562i \(0.640198\pi\)
\(264\) 3.70520e7 0.123936
\(265\) 1.73534e8 0.572829
\(266\) 1.20105e7 0.0391270
\(267\) −3.53895e7 −0.113785
\(268\) 1.03122e8 0.327251
\(269\) −2.42927e8 −0.760927 −0.380464 0.924796i \(-0.624236\pi\)
−0.380464 + 0.924796i \(0.624236\pi\)
\(270\) −1.96830e7 −0.0608581
\(271\) −2.79870e8 −0.854209 −0.427104 0.904202i \(-0.640466\pi\)
−0.427104 + 0.904202i \(0.640466\pi\)
\(272\) −7.25650e7 −0.218643
\(273\) −7.18326e7 −0.213674
\(274\) 7.24139e7 0.212664
\(275\) 4.18791e7 0.121432
\(276\) 1.05564e8 0.302229
\(277\) −4.13906e8 −1.17010 −0.585049 0.810998i \(-0.698925\pi\)
−0.585049 + 0.810998i \(0.698925\pi\)
\(278\) −3.52185e8 −0.983137
\(279\) 1.38075e7 0.0380627
\(280\) −1.40085e7 −0.0381363
\(281\) −1.15682e8 −0.311025 −0.155512 0.987834i \(-0.549703\pi\)
−0.155512 + 0.987834i \(0.549703\pi\)
\(282\) −2.01125e8 −0.534065
\(283\) 4.76235e8 1.24902 0.624509 0.781018i \(-0.285299\pi\)
0.624509 + 0.781018i \(0.285299\pi\)
\(284\) −3.68573e8 −0.954793
\(285\) −2.31491e7 −0.0592349
\(286\) −2.60624e8 −0.658769
\(287\) 3.62472e6 0.00905081
\(288\) 2.38879e7 0.0589256
\(289\) −9.64797e7 −0.235122
\(290\) 5.04277e7 0.121416
\(291\) 2.40332e8 0.571724
\(292\) 1.02615e8 0.241195
\(293\) 3.99799e8 0.928551 0.464275 0.885691i \(-0.346315\pi\)
0.464275 + 0.885691i \(0.346315\pi\)
\(294\) −1.67537e8 −0.384498
\(295\) −1.55362e8 −0.352344
\(296\) −1.92040e8 −0.430399
\(297\) 5.27556e7 0.116848
\(298\) 1.92474e8 0.421322
\(299\) −7.42541e8 −1.60647
\(300\) 2.70000e7 0.0577350
\(301\) −2.18863e6 −0.00462584
\(302\) −3.43124e8 −0.716848
\(303\) −5.12868e8 −1.05915
\(304\) 2.80945e7 0.0573539
\(305\) −2.47608e8 −0.499707
\(306\) −1.03320e8 −0.206139
\(307\) −3.94980e8 −0.779096 −0.389548 0.921006i \(-0.627369\pi\)
−0.389548 + 0.921006i \(0.627369\pi\)
\(308\) 3.75464e7 0.0732219
\(309\) 2.85967e7 0.0551394
\(310\) −1.89403e7 −0.0361094
\(311\) 2.04626e7 0.0385744 0.0192872 0.999814i \(-0.493860\pi\)
0.0192872 + 0.999814i \(0.493860\pi\)
\(312\) −1.68028e8 −0.313213
\(313\) −8.08100e7 −0.148957 −0.0744784 0.997223i \(-0.523729\pi\)
−0.0744784 + 0.997223i \(0.523729\pi\)
\(314\) 2.03343e8 0.370659
\(315\) −1.99457e7 −0.0359552
\(316\) −1.42389e8 −0.253846
\(317\) 1.03352e9 1.82226 0.911132 0.412115i \(-0.135210\pi\)
0.911132 + 0.412115i \(0.135210\pi\)
\(318\) −2.99867e8 −0.522919
\(319\) −1.35159e8 −0.233120
\(320\) −3.27680e7 −0.0559017
\(321\) −2.15656e8 −0.363910
\(322\) 1.06973e8 0.178558
\(323\) −1.21514e8 −0.200641
\(324\) 3.40122e7 0.0555556
\(325\) −1.89918e8 −0.306885
\(326\) 1.05074e8 0.167970
\(327\) −2.57863e8 −0.407823
\(328\) 8.47877e6 0.0132671
\(329\) −2.03809e8 −0.315528
\(330\) −7.23671e7 −0.110852
\(331\) 2.46710e8 0.373929 0.186965 0.982367i \(-0.440135\pi\)
0.186965 + 0.982367i \(0.440135\pi\)
\(332\) −1.11010e8 −0.166486
\(333\) −2.73432e8 −0.405784
\(334\) 1.33471e8 0.196008
\(335\) −2.01411e8 −0.292702
\(336\) 2.42067e7 0.0348135
\(337\) −7.59669e8 −1.08123 −0.540617 0.841269i \(-0.681809\pi\)
−0.540617 + 0.841269i \(0.681809\pi\)
\(338\) 6.79919e8 0.957743
\(339\) 4.52886e8 0.631379
\(340\) 1.41729e8 0.195560
\(341\) 5.07650e7 0.0693304
\(342\) 4.00017e7 0.0540738
\(343\) −3.50032e8 −0.468358
\(344\) −5.11955e6 −0.00678074
\(345\) −2.06180e8 −0.270322
\(346\) −2.51265e8 −0.326111
\(347\) 1.31031e9 1.68353 0.841765 0.539844i \(-0.181517\pi\)
0.841765 + 0.539844i \(0.181517\pi\)
\(348\) −8.71390e7 −0.110837
\(349\) 9.60572e7 0.120960 0.0604799 0.998169i \(-0.480737\pi\)
0.0604799 + 0.998169i \(0.480737\pi\)
\(350\) 2.73603e7 0.0341101
\(351\) −2.39242e8 −0.295300
\(352\) 8.78269e7 0.107332
\(353\) −5.64318e8 −0.682830 −0.341415 0.939913i \(-0.610906\pi\)
−0.341415 + 0.939913i \(0.610906\pi\)
\(354\) 2.68465e8 0.321645
\(355\) 7.19869e8 0.853993
\(356\) −8.38862e7 −0.0985407
\(357\) −1.04699e8 −0.121788
\(358\) −2.85263e8 −0.328591
\(359\) 2.54090e8 0.289839 0.144919 0.989443i \(-0.453708\pi\)
0.144919 + 0.989443i \(0.453708\pi\)
\(360\) −4.66560e7 −0.0527046
\(361\) 4.70459e7 0.0526316
\(362\) −7.09557e8 −0.786154
\(363\) −3.32191e8 −0.364514
\(364\) −1.70270e8 −0.185047
\(365\) −2.00419e8 −0.215732
\(366\) 4.27867e8 0.456168
\(367\) 4.56848e8 0.482437 0.241219 0.970471i \(-0.422453\pi\)
0.241219 + 0.970471i \(0.422453\pi\)
\(368\) 2.50227e8 0.261738
\(369\) 1.20723e7 0.0125083
\(370\) 3.75078e8 0.384960
\(371\) −3.03869e8 −0.308943
\(372\) 3.27288e7 0.0329633
\(373\) −3.12198e8 −0.311494 −0.155747 0.987797i \(-0.549778\pi\)
−0.155747 + 0.987797i \(0.549778\pi\)
\(374\) −3.79870e8 −0.375478
\(375\) −5.27344e7 −0.0516398
\(376\) −4.76741e8 −0.462514
\(377\) 6.12937e8 0.589143
\(378\) 3.44661e7 0.0328225
\(379\) −2.04152e8 −0.192626 −0.0963132 0.995351i \(-0.530705\pi\)
−0.0963132 + 0.995351i \(0.530705\pi\)
\(380\) −5.48720e7 −0.0512989
\(381\) −3.97995e8 −0.368672
\(382\) 3.46602e8 0.318134
\(383\) 7.91439e8 0.719816 0.359908 0.932988i \(-0.382808\pi\)
0.359908 + 0.932988i \(0.382808\pi\)
\(384\) 5.66231e7 0.0510310
\(385\) −7.33329e7 −0.0654917
\(386\) −2.61755e8 −0.231654
\(387\) −7.28936e6 −0.00639295
\(388\) 5.69677e8 0.495128
\(389\) 3.17457e8 0.273439 0.136720 0.990610i \(-0.456344\pi\)
0.136720 + 0.990610i \(0.456344\pi\)
\(390\) 3.28179e8 0.280146
\(391\) −1.08228e9 −0.915635
\(392\) −3.97124e8 −0.332985
\(393\) −2.85141e7 −0.0236966
\(394\) 2.01978e8 0.166367
\(395\) 2.78103e8 0.227047
\(396\) 1.25050e8 0.101193
\(397\) 1.11437e9 0.893847 0.446924 0.894572i \(-0.352520\pi\)
0.446924 + 0.894572i \(0.352520\pi\)
\(398\) −1.42262e9 −1.13110
\(399\) 4.05355e7 0.0319470
\(400\) 6.40000e7 0.0500000
\(401\) 1.04008e9 0.805492 0.402746 0.915312i \(-0.368056\pi\)
0.402746 + 0.915312i \(0.368056\pi\)
\(402\) 3.48038e8 0.267199
\(403\) −2.30215e8 −0.175213
\(404\) −1.21569e9 −0.917249
\(405\) −6.64301e7 −0.0496904
\(406\) −8.83019e7 −0.0654831
\(407\) −1.00531e9 −0.739127
\(408\) −2.44907e8 −0.178521
\(409\) 2.25210e8 0.162763 0.0813816 0.996683i \(-0.474067\pi\)
0.0813816 + 0.996683i \(0.474067\pi\)
\(410\) −1.65601e7 −0.0118664
\(411\) 2.44397e8 0.173640
\(412\) 6.77849e7 0.0477521
\(413\) 2.72048e8 0.190029
\(414\) 3.56280e8 0.246769
\(415\) 2.16816e8 0.148910
\(416\) −3.98287e8 −0.271250
\(417\) −1.18862e9 −0.802728
\(418\) 1.47071e8 0.0984943
\(419\) 9.08227e8 0.603178 0.301589 0.953438i \(-0.402483\pi\)
0.301589 + 0.953438i \(0.402483\pi\)
\(420\) −4.72786e7 −0.0311381
\(421\) −1.47524e8 −0.0963554 −0.0481777 0.998839i \(-0.515341\pi\)
−0.0481777 + 0.998839i \(0.515341\pi\)
\(422\) −3.44124e8 −0.222906
\(423\) −6.78797e8 −0.436062
\(424\) −7.10797e8 −0.452861
\(425\) −2.76814e8 −0.174915
\(426\) −1.24393e9 −0.779585
\(427\) 4.33576e8 0.269506
\(428\) −5.11184e8 −0.315155
\(429\) −8.79606e8 −0.537883
\(430\) 9.99912e6 0.00606488
\(431\) −7.39478e7 −0.0444892 −0.0222446 0.999753i \(-0.507081\pi\)
−0.0222446 + 0.999753i \(0.507081\pi\)
\(432\) 8.06216e7 0.0481125
\(433\) 1.77854e9 1.05282 0.526412 0.850230i \(-0.323537\pi\)
0.526412 + 0.850230i \(0.323537\pi\)
\(434\) 3.31656e7 0.0194748
\(435\) 1.70193e8 0.0991358
\(436\) −6.11230e8 −0.353185
\(437\) 4.19020e8 0.240187
\(438\) 3.46324e8 0.196935
\(439\) 3.52368e9 1.98779 0.993894 0.110335i \(-0.0351923\pi\)
0.993894 + 0.110335i \(0.0351923\pi\)
\(440\) −1.71537e8 −0.0960004
\(441\) −5.65437e8 −0.313942
\(442\) 1.72268e9 0.948912
\(443\) −3.20313e9 −1.75050 −0.875248 0.483674i \(-0.839302\pi\)
−0.875248 + 0.483674i \(0.839302\pi\)
\(444\) −6.48135e8 −0.351419
\(445\) 1.63840e8 0.0881375
\(446\) 1.02411e9 0.546607
\(447\) 6.49598e8 0.344008
\(448\) 5.73787e7 0.0301494
\(449\) −1.12003e8 −0.0583939 −0.0291969 0.999574i \(-0.509295\pi\)
−0.0291969 + 0.999574i \(0.509295\pi\)
\(450\) 9.11250e7 0.0471405
\(451\) 4.43854e7 0.0227836
\(452\) 1.07351e9 0.546790
\(453\) −1.15804e9 −0.585304
\(454\) 7.06184e8 0.354179
\(455\) 3.32558e8 0.165511
\(456\) 9.48188e7 0.0468293
\(457\) −1.64644e9 −0.806937 −0.403469 0.914993i \(-0.632196\pi\)
−0.403469 + 0.914993i \(0.632196\pi\)
\(458\) 1.87802e7 0.00913420
\(459\) −3.48705e8 −0.168312
\(460\) −4.88724e8 −0.234105
\(461\) 2.01399e8 0.0957422 0.0478711 0.998854i \(-0.484756\pi\)
0.0478711 + 0.998854i \(0.484756\pi\)
\(462\) 1.26719e8 0.0597854
\(463\) −7.61645e8 −0.356631 −0.178316 0.983973i \(-0.557065\pi\)
−0.178316 + 0.983973i \(0.557065\pi\)
\(464\) −2.06552e8 −0.0959878
\(465\) −6.39235e7 −0.0294832
\(466\) −1.64850e9 −0.754637
\(467\) −3.99968e9 −1.81726 −0.908628 0.417606i \(-0.862869\pi\)
−0.908628 + 0.417606i \(0.862869\pi\)
\(468\) −5.67093e8 −0.255737
\(469\) 3.52682e8 0.157862
\(470\) 9.31134e8 0.413685
\(471\) 6.86282e8 0.302642
\(472\) 6.36361e8 0.278552
\(473\) −2.68003e7 −0.0116446
\(474\) −4.80562e8 −0.207265
\(475\) 1.07172e8 0.0458831
\(476\) −2.48175e8 −0.105471
\(477\) −1.01205e9 −0.426962
\(478\) −1.15567e9 −0.483990
\(479\) −6.32298e7 −0.0262874 −0.0131437 0.999914i \(-0.504184\pi\)
−0.0131437 + 0.999914i \(0.504184\pi\)
\(480\) −1.10592e8 −0.0456435
\(481\) 4.55899e9 1.86793
\(482\) 1.10494e9 0.449443
\(483\) 3.61034e8 0.145792
\(484\) −7.87415e8 −0.315678
\(485\) −1.11265e9 −0.442856
\(486\) 1.14791e8 0.0453609
\(487\) −5.49748e8 −0.215681 −0.107841 0.994168i \(-0.534394\pi\)
−0.107841 + 0.994168i \(0.534394\pi\)
\(488\) 1.01420e9 0.395053
\(489\) 3.54624e8 0.137147
\(490\) 7.75633e8 0.297831
\(491\) −1.91574e9 −0.730384 −0.365192 0.930932i \(-0.618997\pi\)
−0.365192 + 0.930932i \(0.618997\pi\)
\(492\) 2.86159e7 0.0108325
\(493\) 8.93380e8 0.335793
\(494\) −6.66957e8 −0.248916
\(495\) −2.44239e8 −0.0905101
\(496\) 7.75794e7 0.0285470
\(497\) −1.26053e9 −0.460582
\(498\) −3.74659e8 −0.135936
\(499\) −3.40227e9 −1.22579 −0.612897 0.790163i \(-0.709996\pi\)
−0.612897 + 0.790163i \(0.709996\pi\)
\(500\) −1.25000e8 −0.0447214
\(501\) 4.50465e8 0.160040
\(502\) −1.15962e9 −0.409121
\(503\) 2.78875e9 0.977061 0.488530 0.872547i \(-0.337533\pi\)
0.488530 + 0.872547i \(0.337533\pi\)
\(504\) 8.16975e7 0.0284251
\(505\) 2.37439e9 0.820412
\(506\) 1.30991e9 0.449484
\(507\) 2.29473e9 0.781994
\(508\) −9.43395e8 −0.319279
\(509\) 4.26477e9 1.43345 0.716726 0.697355i \(-0.245640\pi\)
0.716726 + 0.697355i \(0.245640\pi\)
\(510\) 4.78334e8 0.159674
\(511\) 3.50946e8 0.116350
\(512\) 1.34218e8 0.0441942
\(513\) 1.35006e8 0.0441511
\(514\) 4.73463e8 0.153786
\(515\) −1.32392e8 −0.0427108
\(516\) −1.72785e7 −0.00553645
\(517\) −2.49569e9 −0.794279
\(518\) −6.56785e8 −0.207620
\(519\) −8.48019e8 −0.266269
\(520\) 7.77905e8 0.242614
\(521\) −6.50138e8 −0.201406 −0.100703 0.994917i \(-0.532109\pi\)
−0.100703 + 0.994917i \(0.532109\pi\)
\(522\) −2.94094e8 −0.0904981
\(523\) 5.09320e9 1.55681 0.778404 0.627764i \(-0.216030\pi\)
0.778404 + 0.627764i \(0.216030\pi\)
\(524\) −6.75889e7 −0.0205218
\(525\) 9.23411e7 0.0278508
\(526\) −2.01244e9 −0.602939
\(527\) −3.35547e8 −0.0998658
\(528\) 2.96416e8 0.0876360
\(529\) 3.27225e8 0.0961061
\(530\) 1.38828e9 0.405051
\(531\) 9.06069e8 0.262622
\(532\) 9.60842e7 0.0276669
\(533\) −2.01284e8 −0.0575790
\(534\) −2.83116e8 −0.0804582
\(535\) 9.98407e8 0.281883
\(536\) 8.24978e8 0.231401
\(537\) −9.62764e8 −0.268293
\(538\) −1.94342e9 −0.538057
\(539\) −2.07890e9 −0.571838
\(540\) −1.57464e8 −0.0430331
\(541\) 5.43521e9 1.47580 0.737898 0.674912i \(-0.235819\pi\)
0.737898 + 0.674912i \(0.235819\pi\)
\(542\) −2.23896e9 −0.604017
\(543\) −2.39476e9 −0.641892
\(544\) −5.80520e8 −0.154604
\(545\) 1.19381e9 0.315898
\(546\) −5.74661e8 −0.151091
\(547\) 2.16765e9 0.566283 0.283141 0.959078i \(-0.408623\pi\)
0.283141 + 0.959078i \(0.408623\pi\)
\(548\) 5.79311e8 0.150376
\(549\) 1.44405e9 0.372459
\(550\) 3.35033e8 0.0858654
\(551\) −3.45883e8 −0.0880844
\(552\) 8.44515e8 0.213708
\(553\) −4.86975e8 −0.122453
\(554\) −3.31125e9 −0.827384
\(555\) 1.26589e9 0.314319
\(556\) −2.81748e9 −0.695183
\(557\) 1.50603e8 0.0369268 0.0184634 0.999830i \(-0.494123\pi\)
0.0184634 + 0.999830i \(0.494123\pi\)
\(558\) 1.10460e8 0.0269144
\(559\) 1.21537e8 0.0294284
\(560\) −1.12068e8 −0.0269664
\(561\) −1.28206e9 −0.306576
\(562\) −9.25459e8 −0.219928
\(563\) 4.26461e9 1.00716 0.503581 0.863948i \(-0.332015\pi\)
0.503581 + 0.863948i \(0.332015\pi\)
\(564\) −1.60900e9 −0.377641
\(565\) −2.09670e9 −0.489064
\(566\) 3.80988e9 0.883189
\(567\) 1.16323e8 0.0267994
\(568\) −2.94858e9 −0.675141
\(569\) 5.54924e9 1.26282 0.631409 0.775450i \(-0.282477\pi\)
0.631409 + 0.775450i \(0.282477\pi\)
\(570\) −1.85193e8 −0.0418854
\(571\) −6.70520e9 −1.50725 −0.753625 0.657304i \(-0.771697\pi\)
−0.753625 + 0.657304i \(0.771697\pi\)
\(572\) −2.08499e9 −0.465820
\(573\) 1.16978e9 0.259755
\(574\) 2.89977e7 0.00639989
\(575\) 9.54539e8 0.209390
\(576\) 1.91103e8 0.0416667
\(577\) 2.69994e9 0.585112 0.292556 0.956248i \(-0.405494\pi\)
0.292556 + 0.956248i \(0.405494\pi\)
\(578\) −7.71838e8 −0.166256
\(579\) −8.83425e8 −0.189145
\(580\) 4.03421e8 0.0858541
\(581\) −3.79659e8 −0.0803113
\(582\) 1.92266e9 0.404270
\(583\) −3.72094e9 −0.777702
\(584\) 8.20917e8 0.170551
\(585\) 1.10760e9 0.228738
\(586\) 3.19840e9 0.656584
\(587\) 2.93647e9 0.599229 0.299615 0.954060i \(-0.403142\pi\)
0.299615 + 0.954060i \(0.403142\pi\)
\(588\) −1.34029e9 −0.271881
\(589\) 1.29911e8 0.0261965
\(590\) −1.24289e9 −0.249145
\(591\) 6.81676e8 0.135838
\(592\) −1.53632e9 −0.304338
\(593\) −1.37844e9 −0.271454 −0.135727 0.990746i \(-0.543337\pi\)
−0.135727 + 0.990746i \(0.543337\pi\)
\(594\) 4.22045e8 0.0826240
\(595\) 4.84717e8 0.0943363
\(596\) 1.53979e9 0.297920
\(597\) −4.80136e9 −0.923536
\(598\) −5.94033e9 −1.13594
\(599\) −3.76637e8 −0.0716027 −0.0358013 0.999359i \(-0.511398\pi\)
−0.0358013 + 0.999359i \(0.511398\pi\)
\(600\) 2.16000e8 0.0408248
\(601\) 2.11400e9 0.397232 0.198616 0.980077i \(-0.436355\pi\)
0.198616 + 0.980077i \(0.436355\pi\)
\(602\) −1.75091e7 −0.00327096
\(603\) 1.17463e9 0.218167
\(604\) −2.74499e9 −0.506888
\(605\) 1.53792e9 0.282351
\(606\) −4.10295e9 −0.748931
\(607\) −8.91110e9 −1.61723 −0.808613 0.588340i \(-0.799782\pi\)
−0.808613 + 0.588340i \(0.799782\pi\)
\(608\) 2.24756e8 0.0405554
\(609\) −2.98019e8 −0.0534667
\(610\) −1.98086e9 −0.353346
\(611\) 1.13177e10 2.00731
\(612\) −8.26561e8 −0.145762
\(613\) 6.31909e8 0.110801 0.0554004 0.998464i \(-0.482356\pi\)
0.0554004 + 0.998464i \(0.482356\pi\)
\(614\) −3.15984e9 −0.550904
\(615\) −5.58903e7 −0.00968889
\(616\) 3.00371e8 0.0517757
\(617\) −2.14564e9 −0.367755 −0.183877 0.982949i \(-0.558865\pi\)
−0.183877 + 0.982949i \(0.558865\pi\)
\(618\) 2.28774e8 0.0389894
\(619\) 1.84414e9 0.312519 0.156259 0.987716i \(-0.450056\pi\)
0.156259 + 0.987716i \(0.450056\pi\)
\(620\) −1.51522e8 −0.0255332
\(621\) 1.20244e9 0.201486
\(622\) 1.63701e8 0.0272762
\(623\) −2.86894e8 −0.0475350
\(624\) −1.34422e9 −0.221475
\(625\) 2.44141e8 0.0400000
\(626\) −6.46480e8 −0.105328
\(627\) 4.96366e8 0.0804203
\(628\) 1.62674e9 0.262096
\(629\) 6.64491e9 1.06466
\(630\) −1.59565e8 −0.0254242
\(631\) 6.58147e9 1.04285 0.521423 0.853299i \(-0.325402\pi\)
0.521423 + 0.853299i \(0.325402\pi\)
\(632\) −1.13911e9 −0.179496
\(633\) −1.16142e9 −0.182002
\(634\) 8.26816e9 1.28854
\(635\) 1.84257e9 0.285572
\(636\) −2.39894e9 −0.369760
\(637\) 9.42765e9 1.44516
\(638\) −1.08128e9 −0.164841
\(639\) −4.19828e9 −0.636529
\(640\) −2.62144e8 −0.0395285
\(641\) 9.54018e9 1.43072 0.715358 0.698758i \(-0.246264\pi\)
0.715358 + 0.698758i \(0.246264\pi\)
\(642\) −1.72525e9 −0.257323
\(643\) 1.86320e9 0.276388 0.138194 0.990405i \(-0.455870\pi\)
0.138194 + 0.990405i \(0.455870\pi\)
\(644\) 8.55785e8 0.126260
\(645\) 3.37470e7 0.00495196
\(646\) −9.72116e8 −0.141874
\(647\) 9.48581e9 1.37692 0.688461 0.725273i \(-0.258287\pi\)
0.688461 + 0.725273i \(0.258287\pi\)
\(648\) 2.72098e8 0.0392837
\(649\) 3.33128e9 0.478360
\(650\) −1.51935e9 −0.217000
\(651\) 1.11934e8 0.0159011
\(652\) 8.40590e8 0.118773
\(653\) −6.17298e9 −0.867559 −0.433779 0.901019i \(-0.642820\pi\)
−0.433779 + 0.901019i \(0.642820\pi\)
\(654\) −2.06290e9 −0.288374
\(655\) 1.32010e8 0.0183553
\(656\) 6.78302e7 0.00938122
\(657\) 1.16884e9 0.160797
\(658\) −1.63047e9 −0.223112
\(659\) 5.38863e9 0.733465 0.366733 0.930326i \(-0.380476\pi\)
0.366733 + 0.930326i \(0.380476\pi\)
\(660\) −5.78937e8 −0.0783840
\(661\) 1.44276e10 1.94307 0.971536 0.236891i \(-0.0761284\pi\)
0.971536 + 0.236891i \(0.0761284\pi\)
\(662\) 1.97368e9 0.264408
\(663\) 5.81404e9 0.774784
\(664\) −8.88080e8 −0.117724
\(665\) −1.87664e8 −0.0247461
\(666\) −2.18746e9 −0.286932
\(667\) −3.08065e9 −0.401978
\(668\) 1.06777e9 0.138599
\(669\) 3.45638e9 0.446303
\(670\) −1.61129e9 −0.206972
\(671\) 5.30924e9 0.678427
\(672\) 1.93653e8 0.0246168
\(673\) 1.28631e10 1.62664 0.813321 0.581816i \(-0.197658\pi\)
0.813321 + 0.581816i \(0.197658\pi\)
\(674\) −6.07735e9 −0.764548
\(675\) 3.07547e8 0.0384900
\(676\) 5.43935e9 0.677226
\(677\) 1.04145e10 1.28997 0.644986 0.764195i \(-0.276863\pi\)
0.644986 + 0.764195i \(0.276863\pi\)
\(678\) 3.62309e9 0.446452
\(679\) 1.94832e9 0.238845
\(680\) 1.13383e9 0.138282
\(681\) 2.38337e9 0.289186
\(682\) 4.06120e8 0.0490240
\(683\) 7.78284e9 0.934686 0.467343 0.884076i \(-0.345211\pi\)
0.467343 + 0.884076i \(0.345211\pi\)
\(684\) 3.20014e8 0.0382360
\(685\) −1.13147e9 −0.134501
\(686\) −2.80025e9 −0.331179
\(687\) 6.33831e7 0.00745804
\(688\) −4.09564e7 −0.00479471
\(689\) 1.68742e10 1.96542
\(690\) −1.64944e9 −0.191146
\(691\) −1.41219e10 −1.62824 −0.814122 0.580694i \(-0.802781\pi\)
−0.814122 + 0.580694i \(0.802781\pi\)
\(692\) −2.01012e9 −0.230595
\(693\) 4.27677e8 0.0488146
\(694\) 1.04825e10 1.19044
\(695\) 5.50289e9 0.621790
\(696\) −6.97112e8 −0.0783737
\(697\) −2.93380e8 −0.0328183
\(698\) 7.68458e8 0.0855315
\(699\) −5.56368e9 −0.616158
\(700\) 2.18883e8 0.0241195
\(701\) 4.17672e9 0.457955 0.228977 0.973432i \(-0.426462\pi\)
0.228977 + 0.973432i \(0.426462\pi\)
\(702\) −1.91394e9 −0.208808
\(703\) −2.57266e9 −0.279279
\(704\) 7.02615e8 0.0758950
\(705\) 3.14258e9 0.337772
\(706\) −4.51454e9 −0.482834
\(707\) −4.15770e9 −0.442471
\(708\) 2.14772e9 0.227437
\(709\) −3.27466e9 −0.345068 −0.172534 0.985004i \(-0.555195\pi\)
−0.172534 + 0.985004i \(0.555195\pi\)
\(710\) 5.75895e9 0.603864
\(711\) −1.62190e9 −0.169231
\(712\) −6.71090e8 −0.0696788
\(713\) 1.15707e9 0.119549
\(714\) −8.37591e8 −0.0861169
\(715\) 4.07225e9 0.416642
\(716\) −2.28211e9 −0.232349
\(717\) −3.90039e9 −0.395176
\(718\) 2.03272e9 0.204947
\(719\) 1.34003e10 1.34451 0.672253 0.740322i \(-0.265327\pi\)
0.672253 + 0.740322i \(0.265327\pi\)
\(720\) −3.73248e8 −0.0372678
\(721\) 2.31827e8 0.0230351
\(722\) 3.76367e8 0.0372161
\(723\) 3.72917e9 0.366968
\(724\) −5.67646e9 −0.555895
\(725\) −7.87932e8 −0.0767902
\(726\) −2.65753e9 −0.257750
\(727\) −1.38074e10 −1.33273 −0.666363 0.745627i \(-0.732150\pi\)
−0.666363 + 0.745627i \(0.732150\pi\)
\(728\) −1.36216e9 −0.130848
\(729\) 3.87420e8 0.0370370
\(730\) −1.60335e9 −0.152545
\(731\) 1.77145e8 0.0167733
\(732\) 3.42293e9 0.322559
\(733\) −3.61932e9 −0.339440 −0.169720 0.985492i \(-0.554286\pi\)
−0.169720 + 0.985492i \(0.554286\pi\)
\(734\) 3.65479e9 0.341135
\(735\) 2.61776e9 0.243178
\(736\) 2.00181e9 0.185077
\(737\) 4.31867e9 0.397387
\(738\) 9.65785e7 0.00884470
\(739\) 1.69854e9 0.154817 0.0774087 0.996999i \(-0.475335\pi\)
0.0774087 + 0.996999i \(0.475335\pi\)
\(740\) 3.00063e9 0.272208
\(741\) −2.25098e9 −0.203239
\(742\) −2.43095e9 −0.218455
\(743\) −4.34751e9 −0.388848 −0.194424 0.980918i \(-0.562284\pi\)
−0.194424 + 0.980918i \(0.562284\pi\)
\(744\) 2.61831e8 0.0233085
\(745\) −3.00740e9 −0.266467
\(746\) −2.49759e9 −0.220259
\(747\) −1.26447e9 −0.110991
\(748\) −3.03896e9 −0.265503
\(749\) −1.74827e9 −0.152028
\(750\) −4.21875e8 −0.0365148
\(751\) −1.90331e10 −1.63972 −0.819858 0.572566i \(-0.805948\pi\)
−0.819858 + 0.572566i \(0.805948\pi\)
\(752\) −3.81393e9 −0.327047
\(753\) −3.91371e9 −0.334046
\(754\) 4.90349e9 0.416587
\(755\) 5.36132e9 0.453375
\(756\) 2.75729e8 0.0232090
\(757\) 2.46839e8 0.0206813 0.0103407 0.999947i \(-0.496708\pi\)
0.0103407 + 0.999947i \(0.496708\pi\)
\(758\) −1.63321e9 −0.136207
\(759\) 4.42094e9 0.367002
\(760\) −4.38976e8 −0.0362738
\(761\) −1.69781e10 −1.39651 −0.698255 0.715849i \(-0.746040\pi\)
−0.698255 + 0.715849i \(0.746040\pi\)
\(762\) −3.18396e9 −0.260690
\(763\) −2.09043e9 −0.170373
\(764\) 2.77282e9 0.224954
\(765\) 1.61438e9 0.130374
\(766\) 6.33151e9 0.508987
\(767\) −1.51071e10 −1.20892
\(768\) 4.52985e8 0.0360844
\(769\) 1.96829e10 1.56080 0.780400 0.625281i \(-0.215015\pi\)
0.780400 + 0.625281i \(0.215015\pi\)
\(770\) −5.86663e8 −0.0463096
\(771\) 1.59794e9 0.125565
\(772\) −2.09404e9 −0.163804
\(773\) 6.02137e9 0.468886 0.234443 0.972130i \(-0.424673\pi\)
0.234443 + 0.972130i \(0.424673\pi\)
\(774\) −5.83149e7 −0.00452050
\(775\) 2.95942e8 0.0228376
\(776\) 4.55741e9 0.350108
\(777\) −2.21665e9 −0.169521
\(778\) 2.53966e9 0.193351
\(779\) 1.13586e8 0.00860880
\(780\) 2.62543e9 0.198093
\(781\) −1.54355e10 −1.15942
\(782\) −8.65827e9 −0.647452
\(783\) −9.92568e8 −0.0738914
\(784\) −3.17699e9 −0.235456
\(785\) −3.17723e9 −0.234426
\(786\) −2.28112e8 −0.0167560
\(787\) 1.41731e10 1.03646 0.518230 0.855241i \(-0.326591\pi\)
0.518230 + 0.855241i \(0.326591\pi\)
\(788\) 1.61583e9 0.117639
\(789\) −6.79200e9 −0.492298
\(790\) 2.22482e9 0.160546
\(791\) 3.67144e9 0.263766
\(792\) 1.00040e9 0.0715545
\(793\) −2.40769e10 −1.71453
\(794\) 8.91497e9 0.632045
\(795\) 4.68543e9 0.330723
\(796\) −1.13810e10 −0.799806
\(797\) 2.66961e10 1.86786 0.933928 0.357462i \(-0.116358\pi\)
0.933928 + 0.357462i \(0.116358\pi\)
\(798\) 3.24284e8 0.0225900
\(799\) 1.64960e10 1.14410
\(800\) 5.12000e8 0.0353553
\(801\) −9.55517e8 −0.0656938
\(802\) 8.32063e9 0.569569
\(803\) 4.29741e9 0.292888
\(804\) 2.78430e9 0.188938
\(805\) −1.67146e9 −0.112930
\(806\) −1.84172e9 −0.123894
\(807\) −6.55903e9 −0.439322
\(808\) −9.72550e9 −0.648593
\(809\) 2.58931e10 1.71935 0.859676 0.510839i \(-0.170665\pi\)
0.859676 + 0.510839i \(0.170665\pi\)
\(810\) −5.31441e8 −0.0351364
\(811\) −5.90009e9 −0.388405 −0.194203 0.980961i \(-0.562212\pi\)
−0.194203 + 0.980961i \(0.562212\pi\)
\(812\) −7.06415e8 −0.0463035
\(813\) −7.55649e9 −0.493178
\(814\) −8.04247e9 −0.522641
\(815\) −1.64178e9 −0.106234
\(816\) −1.95926e9 −0.126234
\(817\) −6.85840e7 −0.00439993
\(818\) 1.80168e9 0.115091
\(819\) −1.93948e9 −0.123365
\(820\) −1.32481e8 −0.00839082
\(821\) −6.68653e9 −0.421696 −0.210848 0.977519i \(-0.567623\pi\)
−0.210848 + 0.977519i \(0.567623\pi\)
\(822\) 1.95517e9 0.122782
\(823\) 1.16174e10 0.726454 0.363227 0.931701i \(-0.381675\pi\)
0.363227 + 0.931701i \(0.381675\pi\)
\(824\) 5.42279e8 0.0337658
\(825\) 1.13074e9 0.0701088
\(826\) 2.17638e9 0.134371
\(827\) 1.38198e10 0.849632 0.424816 0.905280i \(-0.360339\pi\)
0.424816 + 0.905280i \(0.360339\pi\)
\(828\) 2.85024e9 0.174492
\(829\) −1.82270e10 −1.11115 −0.555576 0.831465i \(-0.687502\pi\)
−0.555576 + 0.831465i \(0.687502\pi\)
\(830\) 1.73453e9 0.105295
\(831\) −1.11755e10 −0.675557
\(832\) −3.18630e9 −0.191803
\(833\) 1.37412e10 0.823695
\(834\) −9.50899e9 −0.567614
\(835\) −2.08549e9 −0.123967
\(836\) 1.17657e9 0.0696460
\(837\) 3.72802e8 0.0219755
\(838\) 7.26582e9 0.426511
\(839\) 1.50479e10 0.879646 0.439823 0.898084i \(-0.355041\pi\)
0.439823 + 0.898084i \(0.355041\pi\)
\(840\) −3.78229e8 −0.0220180
\(841\) −1.47069e10 −0.852582
\(842\) −1.18019e9 −0.0681336
\(843\) −3.12342e9 −0.179570
\(844\) −2.75299e9 −0.157618
\(845\) −1.06237e10 −0.605730
\(846\) −5.43038e9 −0.308343
\(847\) −2.69299e9 −0.152280
\(848\) −5.68637e9 −0.320221
\(849\) 1.28583e10 0.721121
\(850\) −2.21451e9 −0.123683
\(851\) −2.29137e10 −1.27451
\(852\) −9.95147e9 −0.551250
\(853\) −3.30273e9 −0.182201 −0.0911006 0.995842i \(-0.529038\pi\)
−0.0911006 + 0.995842i \(0.529038\pi\)
\(854\) 3.46861e9 0.190569
\(855\) −6.25026e8 −0.0341993
\(856\) −4.08947e9 −0.222848
\(857\) 1.50756e10 0.818167 0.409083 0.912497i \(-0.365849\pi\)
0.409083 + 0.912497i \(0.365849\pi\)
\(858\) −7.03684e9 −0.380340
\(859\) 3.66462e10 1.97266 0.986331 0.164774i \(-0.0526896\pi\)
0.986331 + 0.164774i \(0.0526896\pi\)
\(860\) 7.99930e7 0.00428852
\(861\) 9.78674e7 0.00522549
\(862\) −5.91583e8 −0.0314586
\(863\) 1.85096e10 0.980303 0.490151 0.871637i \(-0.336942\pi\)
0.490151 + 0.871637i \(0.336942\pi\)
\(864\) 6.44973e8 0.0340207
\(865\) 3.92602e9 0.206251
\(866\) 1.42283e10 0.744458
\(867\) −2.60495e9 −0.135748
\(868\) 2.65325e8 0.0137708
\(869\) −5.96311e9 −0.308250
\(870\) 1.36155e9 0.0700996
\(871\) −1.95848e10 −1.00428
\(872\) −4.88984e9 −0.249739
\(873\) 6.48897e9 0.330085
\(874\) 3.35216e9 0.169838
\(875\) −4.27505e8 −0.0215731
\(876\) 2.77059e9 0.139254
\(877\) −1.04554e10 −0.523412 −0.261706 0.965148i \(-0.584285\pi\)
−0.261706 + 0.965148i \(0.584285\pi\)
\(878\) 2.81894e10 1.40558
\(879\) 1.07946e10 0.536099
\(880\) −1.37229e9 −0.0678825
\(881\) 1.60550e10 0.791032 0.395516 0.918459i \(-0.370566\pi\)
0.395516 + 0.918459i \(0.370566\pi\)
\(882\) −4.52349e9 −0.221990
\(883\) 2.54528e9 0.124415 0.0622076 0.998063i \(-0.480186\pi\)
0.0622076 + 0.998063i \(0.480186\pi\)
\(884\) 1.37814e10 0.670982
\(885\) −4.19476e9 −0.203426
\(886\) −2.56250e10 −1.23779
\(887\) −3.47758e10 −1.67319 −0.836594 0.547823i \(-0.815457\pi\)
−0.836594 + 0.547823i \(0.815457\pi\)
\(888\) −5.18508e9 −0.248491
\(889\) −3.22645e9 −0.154017
\(890\) 1.31072e9 0.0623226
\(891\) 1.42440e9 0.0674622
\(892\) 8.19289e9 0.386510
\(893\) −6.38665e9 −0.300119
\(894\) 5.19679e9 0.243250
\(895\) 4.45724e9 0.207819
\(896\) 4.59030e8 0.0213188
\(897\) −2.00486e10 −0.927493
\(898\) −8.96023e8 −0.0412907
\(899\) −9.55115e8 −0.0438426
\(900\) 7.29000e8 0.0333333
\(901\) 2.45948e10 1.12023
\(902\) 3.55083e8 0.0161104
\(903\) −5.90931e7 −0.00267073
\(904\) 8.58806e9 0.386639
\(905\) 1.10868e10 0.497207
\(906\) −9.26435e9 −0.413872
\(907\) −2.23850e9 −0.0996165 −0.0498083 0.998759i \(-0.515861\pi\)
−0.0498083 + 0.998759i \(0.515861\pi\)
\(908\) 5.64947e9 0.250442
\(909\) −1.38474e10 −0.611499
\(910\) 2.66047e9 0.117034
\(911\) −2.43248e10 −1.06595 −0.532973 0.846132i \(-0.678925\pi\)
−0.532973 + 0.846132i \(0.678925\pi\)
\(912\) 7.58551e8 0.0331133
\(913\) −4.64900e9 −0.202168
\(914\) −1.31715e10 −0.570591
\(915\) −6.68541e9 −0.288506
\(916\) 1.50241e8 0.00645885
\(917\) −2.31157e8 −0.00989951
\(918\) −2.78964e9 −0.119014
\(919\) −2.38178e10 −1.01227 −0.506136 0.862454i \(-0.668926\pi\)
−0.506136 + 0.862454i \(0.668926\pi\)
\(920\) −3.90979e9 −0.165537
\(921\) −1.06645e10 −0.449811
\(922\) 1.61119e9 0.0677000
\(923\) 6.99987e10 2.93011
\(924\) 1.01375e9 0.0422747
\(925\) −5.86060e9 −0.243470
\(926\) −6.09316e9 −0.252176
\(927\) 7.72112e8 0.0318347
\(928\) −1.65241e9 −0.0678736
\(929\) −3.58483e9 −0.146695 −0.0733473 0.997306i \(-0.523368\pi\)
−0.0733473 + 0.997306i \(0.523368\pi\)
\(930\) −5.11388e8 −0.0208478
\(931\) −5.32007e9 −0.216069
\(932\) −1.31880e10 −0.533609
\(933\) 5.52489e8 0.0222709
\(934\) −3.19974e10 −1.28499
\(935\) 5.93547e9 0.237473
\(936\) −4.53674e9 −0.180833
\(937\) −1.94222e10 −0.771277 −0.385639 0.922650i \(-0.626019\pi\)
−0.385639 + 0.922650i \(0.626019\pi\)
\(938\) 2.82146e9 0.111626
\(939\) −2.18187e9 −0.0860002
\(940\) 7.44907e9 0.292519
\(941\) 8.46440e9 0.331156 0.165578 0.986197i \(-0.447051\pi\)
0.165578 + 0.986197i \(0.447051\pi\)
\(942\) 5.49025e9 0.214000
\(943\) 1.01166e9 0.0392867
\(944\) 5.09089e9 0.196966
\(945\) −5.38533e8 −0.0207587
\(946\) −2.14402e8 −0.00823399
\(947\) 1.07637e10 0.411848 0.205924 0.978568i \(-0.433980\pi\)
0.205924 + 0.978568i \(0.433980\pi\)
\(948\) −3.84449e9 −0.146558
\(949\) −1.94884e10 −0.740191
\(950\) 8.57375e8 0.0324443
\(951\) 2.79050e10 1.05208
\(952\) −1.98540e9 −0.0745794
\(953\) 3.47302e10 1.29982 0.649908 0.760013i \(-0.274807\pi\)
0.649908 + 0.760013i \(0.274807\pi\)
\(954\) −8.09642e9 −0.301907
\(955\) −5.41566e9 −0.201205
\(956\) −9.24537e9 −0.342233
\(957\) −3.64931e9 −0.134592
\(958\) −5.05839e8 −0.0185880
\(959\) 1.98127e9 0.0725400
\(960\) −8.84736e8 −0.0322749
\(961\) −2.71539e10 −0.986961
\(962\) 3.64719e10 1.32083
\(963\) −5.82271e9 −0.210103
\(964\) 8.83952e9 0.317804
\(965\) 4.08993e9 0.146511
\(966\) 2.88828e9 0.103091
\(967\) −2.78429e9 −0.0990198 −0.0495099 0.998774i \(-0.515766\pi\)
−0.0495099 + 0.998774i \(0.515766\pi\)
\(968\) −6.29932e9 −0.223218
\(969\) −3.28089e9 −0.115840
\(970\) −8.90120e9 −0.313146
\(971\) 1.15571e10 0.405117 0.202558 0.979270i \(-0.435074\pi\)
0.202558 + 0.979270i \(0.435074\pi\)
\(972\) 9.18330e8 0.0320750
\(973\) −9.63589e9 −0.335349
\(974\) −4.39799e9 −0.152510
\(975\) −5.12779e9 −0.177180
\(976\) 8.11362e9 0.279345
\(977\) −2.70102e10 −0.926609 −0.463304 0.886199i \(-0.653336\pi\)
−0.463304 + 0.886199i \(0.653336\pi\)
\(978\) 2.83699e9 0.0969777
\(979\) −3.51308e9 −0.119660
\(980\) 6.20507e9 0.210598
\(981\) −6.96229e9 −0.235456
\(982\) −1.53259e10 −0.516460
\(983\) −4.70303e10 −1.57921 −0.789606 0.613614i \(-0.789715\pi\)
−0.789606 + 0.613614i \(0.789715\pi\)
\(984\) 2.28927e8 0.00765974
\(985\) −3.15591e9 −0.105220
\(986\) 7.14704e9 0.237442
\(987\) −5.50284e9 −0.182170
\(988\) −5.33565e9 −0.176010
\(989\) −6.10852e8 −0.0200793
\(990\) −1.95391e9 −0.0640003
\(991\) −1.93734e10 −0.632336 −0.316168 0.948703i \(-0.602396\pi\)
−0.316168 + 0.948703i \(0.602396\pi\)
\(992\) 6.20636e8 0.0201858
\(993\) 6.66118e9 0.215888
\(994\) −1.00843e10 −0.325681
\(995\) 2.22285e10 0.715368
\(996\) −2.99727e9 −0.0961210
\(997\) −4.62926e10 −1.47938 −0.739689 0.672949i \(-0.765027\pi\)
−0.739689 + 0.672949i \(0.765027\pi\)
\(998\) −2.72182e10 −0.866767
\(999\) −7.38266e9 −0.234279
Display \(a_p\) with \(p\) up to: 50 250 1000 (See \(a_n\) instead) (See \(a_n\) instead) (See \(a_n\) instead) Display \(a_n\) with \(n\) up to: 50 250 1000 (See only \(a_p\)) (See only \(a_p\)) (See only \(a_p\))

Twists

       By twisting character
Char Parity Ord Type Twist Min Dim
1.1 even 1 trivial 570.8.a.b.1.3 4
    
        By twisted newform
Twist Min Dim Char Parity Ord Type
570.8.a.b.1.3 4 1.1 even 1 trivial